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A068566 Numerator of Sum_{k=1..n} 1/(k * 2^k).

%I A068566 #44 Feb 07 2024 01:34:53
%S A068566 1,5,2,131,661,1327,1163,148969,447047,44711,983705,7869871,102309709,
%T A068566 204620705,31972079,32739453941,556571077357,556571247527,
%U A068566 10574855234543,42299423848079,42299425233749,84598851790183
%N A068566 Numerator of Sum_{k=1..n} 1/(k * 2^k).
%C A068566 Sum_{k>=1} 1/(k * 2^k) = log(2).
%C A068566 From _Paul Curtz_, Jun 11 2019: (Start)
%C A068566 (Link) page 9:
%C A068566    T0 =           1/2   =    1/2
%C A068566    T1 =   1/2   + 1/8   =    5/8
%C A068566    T2 =   5/8   + 1/24  =    2/3
%C A068566    T3 =   2/3   + 1/64  =  131/192
%C A068566    T4 = 131/192 + 1/160 =  661/960
%C A068566   (T5 = 661/960 + 1/384 = 1327/1920)
%C A068566 ... .
%C A068566 a(n)/A068565(n) is the first and the third column.
%C A068566 The denominators of the second column are essentially A036289, A097064 and A134401. (End)
%H A068566 Robert Israel, <a href="/A068566/b068566.txt">Table of n, a(n) for n = 1..1228</a>
%H A068566 Paul Curtz, <a href="http://paul.curtz.free.fr/wp-content/uploads/These_Curtz_1965.pdf">Accélération de la convergence de certaines séries alternées à l'aide des fonctions de sommation</a>, Thèse de 3ème Cycle d'Analyse Numérique, Faculté des Sciences de l'Université de Paris, 4 mai 1965.
%F A068566 From _Peter Bala_, Feb 05 2024: (Start)
%F A068566 Integral_{x = 0..1} x^n/(1 + x)^(n+1) dx = log(2) - Sum_{k = 1..n} 1/(k * 2^k).
%F A068566 Hence a(n) = the numerator of Integral_{x = 0..1} ((1 + x)^n - x^n)/(1 + x)^(n+1) dx.
%F A068566 Integral_{x = 0..1/2} x^n/(1 - x) dx = Integral_{x >= 2} 1/(x^(n+2) - x^(n+1)) dx = log(2) - a(n)/A068565(n). (End)
%p A068566 map(numer, ListTools:-PartialSums([seq(1/k/2^k,k=1..100)])); # _Robert Israel_, Jul 10 2015
%t A068566 Numerator[Accumulate[Table[1/(k 2^k),{k,30}]]] (* _Harvey P. Dale_, May 11 2013 *)
%t A068566 a[n_]:=Log[2]-Hypergeometric2F1[1+n,1+n,2+n,-1]/(1+n);
%t A068566 Numerator[Table[Simplify[a[n]],{n,1,30}]] (* _Gerry Martens_, Aug 06 2015 *)
%o A068566 (PARI) vector(30, n, numerator(sum(k=1,n, 1/(k * 2^k)))) \\ _Michel Marcus_, Aug 07 2015
%o A068566 (Magma) [Numerator( (&+[1/(2^k*k): k in [1..n]]) ): n in [1..30]]; // _G. C. Greubel_, Jun 30 2019
%o A068566 (Sage) [numerator( sum(1/(2^k*k) for k in (1..n)) ) for n in (1..30)] # _G. C. Greubel_, Jun 30 2019
%o A068566 (GAP) List([1..30], n-> NumeratorRat( Sum([1..n], k-> 1/(2^k*k)) ) ) # _G. C. Greubel_, Jun 30 2019
%Y A068566 Cf. A068565.
%Y A068566 Cf. A036289, A097064, A134401.
%K A068566 easy,frac,nonn
%O A068566 1,2
%A A068566 _Benoit Cloitre_, Mar 25 2002